AMC 12 2011b Problem 24

Let $P(z) = z^8 + \left(4\sqrt{3} + 6\right)z^4 - \left(4\sqrt{3} + 7\right)$. What is the minimum perimeter among all the $8$-sided polygons in the complex plane whose vertices are precisely the zeros of $P(z)$?

\textbf{(A)}\ 4\sqrt{3} + 4 \textbf{(B)}\ 8\sqrt{2} \textbf{(C)}\ 3\sqrt{2} + 3\sqrt{6} \textbf{(D)}\ 4\sqrt{2} + 4\sqrt{3} \textbf{(E)}\ 4\sqrt{3} + 6